It is important to note that Roeder’s original analysis proceeds in a different manner than the finite mixture modeling presented here. The technique presented by Roeder first develops a “best” range of scale parameters based on a specific criterion. Roeder then uses fixed scale parameters taken from this range to develop optimal equal-scale Gaussian mixture models.

You can reproduce Roeder’s point estimate for the density by specifying a five-component Gaussian mixture. In addition, use the EQUATE=SCALE option in the MODEL statement and a RESTRICT statement fixing the first component’s scale parameter at 0.9025 (Roeder’s h = 0.95, scale). The combination of these options produces a mixture of five Gaussian components, each with variance 0.9025. The following statements conduct this analysis:

title2 "Five Components, Equal Variances = 0.9025"; ods select DensityPlot; proc fmm data=galaxies; model v = / K=5 equate=scale; restrict int 0 (scale 1) = 0.9025; ods exclude IterHistory OptInfo ComponentInfo; run; ods graphics off;

The output is shown in Figure 37.18 and Figure 37.19.

Figure 37.18: Reproduction of Roeder’s Five-Component Analysis of Galaxy Data

FMM Analysis of Galaxies Data |

Five Components, Equal Variances = 0.9025 |

The FMM Procedure

Model Information | |
---|---|

Data Set | WORK.GALAXIES |

Response Variable | v |

Type of Model | Homogeneous Mixture |

Distribution | Normal |

Components | 5 |

Link Function | Identity |

Estimation Method | Maximum Likelihood |

Fit Statistics | |
---|---|

-2 Log Likelihood | 412.2 |

AIC (smaller is better) | 430.2 |

AICC (smaller is better) | 432.7 |

BIC (smaller is better) | 451.9 |

Pearson Statistic | 82.5549 |

Effective Parameters | 9 |

Effective Components | 5 |

Linear Constraints at Solution | |||
---|---|---|---|

k = 1 | Constraint Active |
||

Variance | = | 0.90 | Yes |

Parameter Estimates for 'Normal' Model | |||||
---|---|---|---|---|---|

Component | Parameter | Estimate | Standard Error | z Value | Pr > |z| |

1 | Intercept | 26.3266 | 0.7778 | 33.85 | <.0001 |

2 | Intercept | 33.0443 | 0.5485 | 60.25 | <.0001 |

3 | Intercept | 9.7101 | 0.3591 | 27.04 | <.0001 |

4 | Intercept | 23.0295 | 0.2294 | 100.38 | <.0001 |

5 | Intercept | 19.7187 | 0.1784 | 110.55 | <.0001 |

1 | Variance | 0.9025 | 0 | ||

2 | Variance | 0.9025 | 0 | ||

3 | Variance | 0.9025 | 0 | ||

4 | Variance | 0.9025 | 0 | ||

5 | Variance | 0.9025 | 0 |

Parameter Estimates for Mixing Probabilities | ||||||
---|---|---|---|---|---|---|

Component | Parameter | Linked Scale | Probability | |||

Estimate | Standard Error | z Value | Pr > |z| | |||

1 | Probability | -2.4739 | 0.7084 | -3.49 | 0.0005 | 0.0397 |

2 | Probability | -2.5544 | 0.6016 | -4.25 | <.0001 | 0.0366 |

3 | Probability | -1.7071 | 0.4141 | -4.12 | <.0001 | 0.0854 |

4 | Probability | -0.2466 | 0.2699 | -0.91 | 0.3609 | 0.3678 |

Figure 37.19: Density Plot for Roeder’s Analysis